Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules

Find the first derivative of the function 𝑦 = βˆ’7π‘₯⁴ ln 6π‘₯⁴.

02:12

Video Transcript

Find the first derivative of the function 𝑦 equals negative seven π‘₯ to the fourth power times the natural log of six π‘₯ to the fourth power.

Here, we have the product of two differentiable functions. The first is negative seven π‘₯ to the fourth power and the second is the natural log of six π‘₯ of the fourth power. We can, therefore, find the first derivative of our function by using the product rule. This says that the derivative of the product of two differentiable functions, 𝑒 and 𝑣, is 𝑒 times d𝑣 by dπ‘₯ plus 𝑣 times d𝑒 by dπ‘₯. If we let 𝑒 be equal to negative seven π‘₯ to the fourth power, then d𝑒 by dπ‘₯ is equal to four times negative seven π‘₯ cubed. That’s negative 28π‘₯ cubed. We then let 𝑣 be equal to the natural log of six π‘₯ to the fourth power.

So how do we obtain d𝑣 by dπ‘₯? Well, we can do one of two things. We could spot that we have a composite function and use the chain rule to find its derivative. Alternatively, we can quote the general result for the derivative of the logarithm of some function 𝑓 of π‘₯. This says that if 𝑦 is equal to the natural log of this function 𝑓 of π‘₯, then d𝑦 by dπ‘₯ is equal to 𝑓 prime of π‘₯, the derivative of that function, divided by the original function 𝑓 of π‘₯. In this case, 𝑓 of π‘₯ is equal to six π‘₯ to the fourth power. So 𝑓 prime of π‘₯, the derivative of this, is four times six π‘₯ cubed, which is 24π‘₯ cubed. d𝑣 by dπ‘₯ is, therefore, 24π‘₯ cubed, divided by the original function, six π‘₯ to the fourth power.

Well, this simplifies quite nicely to four over π‘₯. And we can now substitute everything we have into the formula for the product rule. d𝑦 by dπ‘₯ is 𝑒 times d𝑣 by dπ‘₯. That’s negative seven π‘₯ to the fourth power times four over π‘₯ plus 𝑣 times d𝑒 by dπ‘₯. That’s the natural log of six π‘₯ to the fourth power times negative 28π‘₯ cubed. We simplify, and then we factor negative 28π‘₯ cubed. And we obtained d𝑦 by dπ‘₯ to be negative 28π‘₯ cubed times the natural logarithm six π‘₯ to the fourth power plus one.

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